Dynamic Semiparametric Factor Models in RiskNeutral Density Estimation

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SFB 649 Discussion Paper 2008-038

Dynamic Semiparametric Factor Models in Risk

Neutral Density Estimation

Enzo Giacomini*

Wolfgang Härdle*

Volker Krätschmer**

* Humboldt-Universität zu Berlin, Germany

** Technische Universität Berlin, Germany

This research was supported by the Deutsche

Forschungsgemeinschaft through the SFB 649 "Economic Risk".

http://sfb649.wiwi.hu-berlin.de ISSN 1860-5664

S FB

6 4 9

E C O N O M I C

R I S K

B E R L I N

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Dynamic Semiparametric Factor Models in Risk Neutral Density Estimation

Enzo Giacomini

^a,

∗ , Wolfgang Karl H¨ ardle

^a

, Volker Kr¨ atschmer

^a,b

aHumboldt-Universit¨at zu Berlin,

CASE - Center for Applied Statistics and Economics Spandauerstr. 1, 10178 Berlin, Germany

bTechnische Universit¨at Berlin, Institute of Mathematics Straße des 17. Juni 136, 10623 Berlin, Germany

Abstract

Dimension reduction techniques for functional data analysis model and approximate smooth random functions by lower dimensional objects. In many applications the focus of interest lies not only in dimension reduction but also in the dynamic behaviour of the lower dimensional objects. The most prominent dimension reduction technique - functional principal components analysis - however, does not model time dependences embedded in functional data. In this paper we use dynamic semiparametric factor models (DSFM) to reduce dimensionality and analyse the dynamic structure of unknown random functions by means of inference based on their lower dimensional representation. We apply DSFM to estimate the dynamic structure of risk neutral densities implied by prices of option on the DAX stock index.

Key words: dynamic factor models, dimension reduction, risk neutral density AMS classification: 62G08, 62M10, 62P05

JEL classification: C14, C32, G12

1 Introduction

Large datasets containing various samples of high dimensional observations became common in diverse fields of science with advances in measurement and computational techniques. In many applications the data come in curves, i.e.

∗ Corresponding author. Tel. + 49 30 2093 5721, Fax. + 49 30 2093 5649 Email address: giacomini@wiwi.hu-berlin.de(Enzo Giacomini).

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as observations of discretized values of smooth random functions, presenting evident functional structure. In these cases it is natural to perform statistical inference using functional data analysis techniques.

Consider a dataset {(Y_jt, X_jt)}, j = 1, . . . , J_t, t = 1, . . . , T containing noisy samples of a real valued smooth random functionF ∈L₂(X), X ⊆R^d, d∈N evaluated at unbalanced design points Xjt as

Y_jt=F_t(X_jt) +ε_jt (1.1)

where ε_jt denotes unknown zero-mean error terms and {F_t} are i.i.d. realizations of F. Each sample S_t = {(Y_jt, X_jt) : j = 1, . . . , J_t}, t = 1, . . . , T may correspond to a different observation on e.g. a different individual, time period or experimental condition.

Examples in biomedicine are measurements across individuals of growth curves or of brain potentials obtained from EEG, see Kneip and Gasser (1992) and Gasser and Kneip (1995). In econometrics data may originate from expen- ditures on commodities across households, Kneip (1994), implied volatilities observed on different trading days, Fengler, Härdle, and Mammen (2007) or measurements on stock liquidity across time, Dähne, Härdle, and Hautsch (2008). Ramsay and Silverman (2005) and Ferraty and Vieu (2006) provide an extensive list of functional datasets encountered in various applications.

A large branch of functional data analysis concentrates on modelling and approximating the random function F by lower dimensional objects. Distribu- tions on function spaces are highly complex objects and dimension reduction techniques present a feasible and interpretable approach to investigate them.

Functional principal components analysis (FPCA), based on the Karhunen- Lo`eve expansion of F, is the most prominent and widely used dimension reduction technique, see Rao (1958), Rice and Silverman (1991) and Ramsay and Dalzell (1991).

Asymptotic results on FPCA have been obtained by Dauxois, Pousse, and Ro- main (1982) for directly observed functional data {F_t(X_jt)}. In cases where functional data is not directly observable a preliminary smoothing is required at eachS_t and FPCA is performed on the smoothed {F^b_t}, see Besse, Cardot, and Ferraty (1997) and Benko, Kneip, and H¨ardle (2008) for recent develop- ments. In practical applications the previous fits may suffer, however, from design-sparseness at each S_t, Cont and da Fonseca (2002) and Fengler et al.

(2007).

In general lines, previous literature combines PCA and dimension reduction with presmoothing for effective dimensional space at fixed time horizon. Var- ious applications, however, involve modelling the dynamics of the unobserved

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random functions and call for dimension reduction techniques that smooth in space and are parametric in time.

In this paper we simultaneously reduce dimensionality avoiding preliminary smoothing and consider the time structure in the sequence{F_t}. We describe F_t as a linear combination of L+ 1<< T unknown smooth functions, called basis functions, m_l ∈L₂(X),l = 0, . . . , L:

F_t(X_jt) =

L

X

l=0

Z_ltm_l(X_jt). (1.2)

Here Z_t = (Z_0t, . . . , Z_Lt)^> is an unobservable random vector taking values on R^L+1. Defining the tuple of functions m = (m₀, . . . , m_L)^>, the dynamic semiparametric factor model (DSFM), Borak, H¨ardle, Mammen, and Park (2008), reads as

Y_jt=Z_t^>m(X_jt) +ε_jt. (1.3)

The basis functions are estimated nonparametrically avoiding specification is- sues, i.e. the shape ofm_lis extracted from the data. AsmandZ_tare estimated simultaneously, the smoothing is transfered directly to m_l, l = 0, . . . , L and design-sparseness becomes secondary. In addition, and essential for investigating dynamics, the random process{Z_t} may be non-stationary.

The form (1.2) is justified when prior knowledge about the available dataset leads to expect some common structure generating eachF_t. Approximating it by a lower dimensional linear combination of common factors is a natural way to handle and describe the unobservable generating mechanism.

In many applications the index t reflects time evolution and the unobservable {Zt}contains the dynamics of{Ft}. Borak et al. (2008) show that under (1.2) the autocorrelation structures of estimated {Z^b_t} and true {Z_t} are asymptotically equivalent. This result implies that no loss is incurred by inferring the dynamic structure from {Z^bt}, i.e. there is no payment for not knowing the true {Z_t}. This fact is essential for investigating cointegration with an- other dynamical systems. An application using financial data is provided by Br¨uggemann, H¨ardle, Mungo, and Trenkler (2008) where the cointegration between estimates {Z^b_t} and macroeconomic time series is analysed.

Note that in the very similar common regressors model from Kneip (1994) the unobservable functions are also considered a linear combination of unspecified common functions as in (1.2) and are estimated from the data. There are however crucial differences between DSFM and common regressors model:

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0.8 0.9

1 1.1 0

0.2 0.4 400

800 1200

0.8 0.9 1 1.1

0 0.1 0.2 0.3 0.4 0.5

Fig. 1. Samples St,t= 1, . . . ,22 of DAX call prices traded on January 2001 (left).

Corresponding unbalanced design {X_jt}(right)

(1) in DSFM{Z_t}is a (non-stationary) random process with autocovariance structure inferable from{Z^b_t}

(2) DSFM is implementable in unbalanced designs

(3) DSFM avoids presmoothing by transferring the smoothing to the basis functions

Hence, in contrast to the dimension reductions techniques FPCA and common regressors, DSFM reduces dimensionality capturing the structural dynamics embedded in the observations.

We apply DSFM on a financial dataset of option prices in order to investigate the dynamics of risk assessments from investors acting in a market. Option prices are a valuable source of information concerning risk assessments from investors about future financial payoffs. The information is summarized in the risk neutral densities q, the continuous counterpart from Arrow-Debreu security prices, see Ait-Sahalia and Lo (1998). Under no arbitrage assumptions the risk neutral densities - corresponding to a risk neutral measure Q - are derived from prices of European call options.

An European call option on St with maturity date T >0 and strikeK >0 is a financial instrument that delivers at time T the random payoff (S_T −K)⁺ where S_t is the price of the underlying asset at time 0 ≤ t ≤ T. Breeden and Litzenberger (1978) show that under no arbitrage assumptions the risk neutral density is obtained from the European call price function C_t through the relation

q_t,T(s_T|s_t) = e^r(T^−t)∂²C_t(s_t, K, T −t)

∂K²

_K=s

T

(1.4) where r >0 is interest rate, see Section (3) for details.

The knowledge about the risk neutral densities delivers (time dependent) risk assessments from investors concerning future payoffs and is essential for appli-

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cations in financial economics, option pricing and risk management as, e.g. estimated risk neutral densities allow less liquid instruments or contracts traded over-the-counter to be priced. From the economic point of view it is natural to investigate the dynamics of the risk neutral densities and the dependence between risk assessments and macroeconomic and financial indicators. This calls for a method that reduces dimensionality and allows for dynamic analysis of the lower dimensionals. Both features are present in DSFM.

We estimate the risk neutral densities based on a dataset containing intraday prices of calls on the German stock index (DAX) from 2001. Each observation consists of a call price Y_jt on a design point X_jt = (κ_jt, τ_jt)^> where j = 1, . . . , J_t denotes the transactions at day t = 1, . . . , T and κ = g(K) is the moneyness, a monotone transformation of strikes K. Exchange regulations impose prespecified values for tradable maturitiesτ =T−tand are responsible for the observed degenerated design, see figure 1.

Following Ait-Sahalia and Lo (1998) and Fengler et al. (2007) the observations are transformed into log-implied volatilitiesY^e_jt = logC_BS⁻¹(Y_jt), see Section (3) for details. These are assumed as discretized noisy values of the log-implied volatility surface evaluated at design points {X_jt}:

Ye_jt= logV_t(X_jt) +ε_jt. (1.5)

where the smooth random functionV ∈ L₂(X),X ⊂R²+, is called the implied volatility surface andε_jt is an error term. The realizations{V_t}are filtered out from the transformed data with DSFM and the risk neutral densities estimated using (1.4) withC_BS(V^b_t) as an estimator forC_t. The dynamics of the estimated {q_b_t,T}is analysed based on the autocorrelation structure of {Z^b_t}.

In the sequel the DSFM estimation method and its asymptotic properties are described (Section 2). In the application part (Section 3), risk neutral densities are defined and estimated from observed prices of European call options on the DAX index (ODAX dataset). Their dynamic structure is analysed by vector autoregressive models.

2 Estimation Method

Consider a dataset{(Yjt, Xjt)}, j = 1, . . . , Jt,t = 1, . . . , T such that

Y_jt=

L

X

l=0

Z_ltm_l(X_jt) +ε_jt (2.1)

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where ε_jt is an unknown error terms with E[ε_jt] = 0 and E[ε²_jt] < ∞ and {ε_jt} are independent. Here Z_t = (Z_0t, . . . , Z_Lt)^> is an unobservable random vector taking values on R^L+1 with Z_0t = 1 and m_l ∈ L₂(X), l = 0, . . . , L are unknown smooth functions, called basis functions, mapping X ⊆ R^d, d ∈ N into real values.

Following Borak et al. (2008), the basis functions are estimated using a series expansion. Defining K normed functions ψ_k : X → R, ^R_Xψ²_k(x)dx = 1, k = 1, . . . , K and a (L+ 1×K) matrix of coefficients Γ = (γl,k), γl,k ∈ R, the tuple of functions m = (m₀, . . . , m_L)^> is approximated by Γ^>ψ where ψ = (ψ₁, . . . , ψ_K)^>. For simplicity of notation we assume that J_t = J does not depend on t. We define the least square estimators as

(Γ,^b Z) = arg^b min

Γ∈G,Z∈Z T

X

t=1 J

X

j=1

nY_jt−Z_t^>Γψ(X_jt)^o² (2.2)

whereG=M(L+ 1, K),Z ={Z ∈ M(T, L+ 1) :Z_0t= 1}andM(a, b) is the set of all (a×b) matrices. The basis functions m are estimated by cm=Γψ.^b Theorem (2.1) gives the asymptotic behaviour of the least squares estimators (Γ,^b Z).^b

Theorem 2.1 Suppose that DSFM holds and that (Γ,^b Z)^b is defined by (2.2).

Under assumptions (A1)-(A8), see Appendix A, it holds for K, J → ∞:

1 T

X

1≤t≤T

Zb_t^>Γ^b −Z_t^>Γ^∗²=O_P(ρ²+δ_K²)

See Borak et al. (2008) for the proof. Note that the model (2.1) is only identi- fiable up to linear transformations. Consider a (L+ 1×L+ 1) regular matrix B = (b_mn) with b_1n = δ_1n and b_m1 = δ_m1 for m, n = 1, . . . , L + 1 where δ_mn=1(m=n). Define Z_t^∗ =B^>Z_t,m^∗ =B⁻¹m, then from (1.2)

F_t(X) =Z_t^>m(X)

=Z_t^>BB⁻¹m(X)

=Z_t^∗^>m^∗(X)

forX ∈ X. On the other hand it is always possible to chose orthonormal basis functions by setting m^∗ =Hm where H is an orthogonal matrix.

Theorem (2.2) states that for anyZ^b_tthere exists a random matrixB such that the autocovariances of {Z^b_t}, Z^e_t= B^>Z^b_t are asymptotically equivalent to the

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autocovariances of the true unobservable {Z_t}. This equivalence is transfered to classical estimation and testing procedures in the context of e.g. vector autoregressive models and in particular justifies inference based on{Z^b_t}when {Zt} is a VAR process. Define for Ht ∈ Z, t = 1, . . . , T: H = T⁻¹^P^T_t=1Ht, H_c,t =H_t−H and H_n,t= (T⁻¹^P^T_s=1H_c,sH_c,s^>)^−1/2H_c,t.

Theorem 2.2 Suppose that DSFM holds and that (Γ,^b Z)^b is defined by (2.2).

Under assumptions (A1)-(A11), see Appendix A, there exists a random matrix B such that for h6= 0, h_d = max(1,1−h), h_u = max(T, T −h) and T → ∞:

1 T

hu

X

t=hd

Ze_c,tZ^e_c,t+h−Z^e_c,t^>− 1 T

hu

X

t=hd

Z_c,t(Z_c,t+h−Z_c,t)^>=OP(T^−1/2)

where Z^e_t =B^>Z^b_t. Moreover

1 T

hu

X

t=hd

Zen,tZe_n,t+h^> − 1 T

hu

X

t=hd

Zn,tZ_n,t+h^> =^OP(T^−1/2)

See Borak et al. (2008) for the proof. Note that, in contrast to FPCA, DSFM does not require stationarity, neither for{Zt}nor{εt}, but only weak assumptions on the average behaviour of Z_t, like being a martingale difference, see Appendix A.

3 Application

Consider a financial market with one risky asset and one riskless bond with constant interest rate r > 0. Let the price of the asset traded on the market be described by the real valued random process {S_t}, t = [0, T], T <∞ on a filtered probability space (Ω,{Ft},P) withFt=σ(S_u, u≤t) andF0 ={∅,Ω}.

Assume further no arbitrage in the financial market in the sense that there exists a (risk neutral) probability measureQ equivalent toP under which the discounted price process{e^−rtS_t} is a martingale.

A European call option at strike K > 0 is a financial instrument that pays Ψ(S_T) = (S_T −K)⁺ at time T. By the risk-neutral valuation principle w.r.t.

Q the price Ct of a European call option at timet is defined to be C_t=e^−r(T^−t)E^Q[Ψ(S_T)|Ft]

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Assuming that {S_t} is a Q-Markov process and denoting the P-density of Q byπ, the price can be rewritten as

C_t(S_t) =e^−r(T^−t)E^Q[Ψ(S_T)|S_t]

=e^−r(T^−t)E^hΨ(S_T)K^tπ(S_t, S_T)|S_tⁱ

where E denotes the expectation under P and K^tπ(S_t, S_T) ^def.= ^E[π|S_E[π|S^t^,S^T^]

t] . The conditional risk neutral distribution of S_T is defined as

Q_S_T|St=st([S_T ≤x])^def.=

Z x

−∞K^tπ(s_t,·) dP_S_T|St=st (3.1) where P_S_T|St=st is the conditional distribution of S_T under S_t = s_t. Special- izing to the following two factor model we assume that the price process has dynamics given by

dS_t=S_tµ(Y_t)dt+S_tσ(Y_t)dW_t¹

hereW¹ is a standardP-Brownian motion andY denotes an external economic factor process modelled by

dYt=g(Yt) +ρdW_t¹+ρdW_t²

where ρ ∈ [−1,1] is some correlation factor, ρ ^def.= √

1−ρ² and W² is a standard P-Brownian motion independent of W¹ under P. Market models of this type are popular in mathematical finance and economics, in particular if Y follows and Ornstein-Uhlenbeck dynamic with mean reversion termg(y) = κ(θ−y) for constantsθ ≥0 andκ >0. Moreover,{St}is aQ-Markov process for any Q, see Hern´andez-Hern´andez and Schied (2007) and the conditional risk neutral distributionQ_S_T|St=st has a density function denoted by q_t,T(·|s_t).

Hence, the call prices C can be expressed as

C_t(s_t, K, T −t) =e^r(T^−t)

Z

(s_T −K)⁺q_t,T(s_T|s_t)ds_T.

We assume that the observed prices in the financial market are built based on the risk neutral valuation principle w.r.t. an unknown risk neutral measure Q. Our interest lies in estimating the conditional risk neutral distribution Q_S_T_|S_t_=s_t, or equivalently its density function q_t,T(·|s_t), implied by Q through (3.1).

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3.1 Estimation

Adapting Breeden and Litzenberger (1978), one can show that the density function q_t,T(·|s_t), called from now on the risk neutral density, is obtained as the second derivative of the call price functionC_t with respect to strike K

qt,T(sT|st) = e^rτ∂²C_t(s_t, K, τ)

∂K²

_K=s

T

(3.2) here τ = T −t is the time to maturity. The price function can be smoothed out of observations from call prices and used in (3.2) to recover state price densities implied on the data.

Following Ait-Sahalia and Lo (1998) smoothing is carried out in the space of implied volatilities. In general the exact form of Ct is hard to determine and based on historical data an estimate of C_t can be obtained by non-parametric regression. In order to cope with the course of dimensionality, Ait-Sahalia and Lo (1998) proposed a semiparametric approach using the Black-Scholes formula evaluated with a non-parametric estimator for the implied volatility in place of the true non-observable volatility.

The implied volatility surface is the function σ_t : R²+ → R+ satisfying for all (K, τ)∈R²+

C_t(s_t, K, τ) =C_BS{s_t, r_t, K, τ, σ_t(K, τ)} (3.3) where CBS(s, r, K, τ, σ) = sΦ(d1) − Ke^−rτΦ(d2) is the Black-Scholes price of Ψ with strike K and maturity τ. Here Φ(x) is the standard normal cdf, d₁(σ) = ⁿlog_K^s+ (r+¹₂σ²)τ^o/(σ√

τ) and d₂(σ) = d₁ −σ√

τ. Note that C_BS(v) = C_BS(s, r, K, τ, v) is a continuous increasing function of v, hence σ_t(K, τ) =C_BS⁻¹{C_t(s_t, K, τ)}.

0.6 0.8 1

1.2 0.2 0.4 0.6 0.8 0.1

0.2 0.3 0.4 0.5

0.6 0.8 1 1.2

0 0.2 0.4 0.6 0.8

Fig. 2. Call and put implied volatilities observed (left), data design (right), ODAX on 20000502

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More generally, the implied volatility surface is considered a smooth random function V ∈ L₂(X) on the space X ⊂ R² of strikes K and maturities τ. Combining (3.2) and (3.3) the functional random variable H ∈ L₂(X) called the risk neutral (RN) surface is defined as

H=e^rτD²C_BS(V) (3.4)

where D^m denotes the m-th derivative with respect to K. Clearly lower dimension objects describingV may be used to analyseH.

A functional dataset containing realizations ofV is however not available, as in an exchange only discretized values ofV_tcorrupted by noise are registered from trades. More specifically, on each dayt= 1, . . . , T there areJtoptions traded, each intra-day trade j = 1, . . . , J_tcorresponds to an observed option price Y_jt at a pair of moneyness and maturitiesX_jt = (κ_jt, τ_jt)^>whereκ=e^rτK/S. The observed implied volatilities at the respective design points are obtained by σ_jt =C_BS⁻¹(Y_jt), see figure (2). For numerical tractability we perform a further transformation into log-implied volatilities Y^e_jt = logσ_jt and use DSFM to model

Yejt= logVt(Xjt) +εjt

where logV_t =Z_t^>m. The implied volatility surface at t is estimated byV^b_t = exp(Z^b_t^>Γψ^b ), recall (2.2). The RN surface is estimated using (3.4) by H^c_t = H(V^b_t) where

H(V) =ϕ(d₂)

( 1 K√

τV + 2d₁DV

V +K√

τ d₁d₂(DV)²

V +K√ τ D²V

)

(3.5) andd₁ =d₁(V) andd₂ =d₂(V). The dynamics of the unobservable sequence of RN surfaces{H_t}implied in the observations{(Y_jt, X_jt)}may be investigated by analysing the lower dimensional {Z^b_t}.

3.2 Results

Here implied volatility and RN surfaces are estimated with DSFM from intraday call prices on the DAX index, i.e. St represents the value of the DAX index at timet. The time ranges from 20010101 to 20020101 (dates are written as year, month, day) corresponding to 253 trading days.

Tensor B-splines, quadratic in τ and cubic in κ directions placed on 8×6 knots, are used for the series estimators ofm. The dimensionLofZ_t is chosen based on

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RV(L) =

PT t=1

PJt

j=1

n

Ye_jt−Z^b_t^>_cm(X_jt)^o²

PT t=1

PJt

j=1(Y^e_jt−Y)²

whereY = (^P^T_t=1^P^J_j=1^t Y^e_jt)/^P^T_t=1J_t. The value 1−RV(L) may be interpreted as the ratio of variation explained by the model to total variation. The order of the splines and number of knots have negligible influence on RV(L), as established by inumerous simulations in Borak et al. (2008). The implied volatility and RN surfaces are estimated with DSFM as in (3.5) with L = 3.

Table 1 shows that the addition of the fourth or fifth basis function results in negligible model fit improvement.

L 1 2 3 4 5

1−RV(L) 0.77 0.97 0.98 0.98 0.98 Table 1

Number of basis functions and explained variation

Figures 3 and 4 depict the estimated loading factors series {Z^b_t} and basis functions mc_l. From (3.5) we obtain a sequence of RN surfaces {H^c_t}, t = 1, . . . ,253, figure 5 shows H^c_t at t corresponding to day 20010710.

Jan01 Apr01 Jul01 Oct01 Jan02

0 0.5 1

z

Fig. 3. Estimated{Zb_lt},l= 1,2,3 (top to bottom)

In a first step we investigate the covariance structure of {Z^bt} by means of VAR analysis. Table 2 presents the parameters from the VAR(2) model fitted on {Z^b_t}. The order 2 is selected based on Akaike (AIC), Schwarz (SC) and Hannan-Quinn (HQ) criteria, see table 3. Moreover the VAR(2) model satisfies stationarity as the roots of the characteristic polynomial lie inside of the unit circle.

A natural issue is to analyse the dependences between {Zt} and the shape of the RN surfaces {H^c_t}. In order to investigate this relation we compute the skewness γ and excess kurtosis η of qb_t,T(·|s_t) across t and maturities τ. Here

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0.7 0.8

0.9 1

1.1 1.2

0.1 0.2

0.3 0.4

0.5

−0.4

−0.2 0 0.2 0.4

0.7 0.8

0.9 1

1.1 1.2

0.1 0.2

0.3 0.4

0.5

−3

−2.5

−2

−1.5

0.7 0.8

0.9

1 1.1

1.2 0.1

0.2 0.3

0.4

−3 0.5

−2

−1 0 1 2

0.7 0.8

0.9

1 1.1

1.2 0.1

0.2 0.3

0.4 0.5

−4

−2 0 2 4

Fig. 4. Estimated basis functions mb_l,l= 0, . . . ,3 clockwise

0.85 0.9 0.95 1 1.05 1.1

0.05 0.1

0.15 0

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 10⁻³

Fig. 5. Estimated RN surface, Hb_tatt corresponding to day 20010710

VAR(2)

const Zb1,t−1 Zb1,t−2 Zb2,t−1 Zb2,t−2 Zb3,t−1 Zb3,t−2

Zb_1t 0.01 1.09 -0.16 0.10 -0.36 0.32 -0.23 Zb2t 0.01 -0.27 0.26 0.31 0.12 -1.14 0.33 Zb_3t 0.01 -0.08 0.62 -0.05 -0.04 0.41 0.35 Table 2

Estimated parameters for the VAR(2) model on{Zbt}

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order AIC SC HQ 1 -11.03 -10.99 -11.01 2 -15.71 -15.54* -15.64 3 -15.77* -15.46 -15.64*

4 -15.76 -15.32 -15.58 5 -15.72 -15.16 -15.45 Table 3

Lag selection criteria for VAR models on{Zbt}

0 50 100 150 200 250

−1.5 0 1.5

0 50 100 150 200 250

0.4 0.8

0 50 100 150 200 250

−1.5 0 1.5

0 50 100 150 200 250

−0.1 0.1

Fig. 6. Left: RN excess kurtosis {η_t}, τ = 18 (top), {Zb_1t} (bottom) Right: RN skewness{γ_t},τ = 18 (top), {Zb2t}(bottom)

qb_t,T(·|s_t) = H^c_t(·, τ). Figure 6 displays skewness {γ_t} and excess kurtosis {η_t} associated with q_b_t,T for maturity τ = 18 days together with {Z^b_1t} and {Z^b_3t} and motivates the investigation of their joint autocovariance structure.

The dynamic structure of the pairs {(Z^b_1t, η_t)} and {(Z^b_3t, γ_t)} for τ = 18 is modelled by VAR(2) models. The choice of the VAR order is again based on AIC, SC, and HQ selection criteria. Portmanteau and LM tests on VAR residuals reject autocorrelations up to lag 12 and the roots of the characteristic polynomial lie inside of the unit circle.

VAR(2)

const Zb1,t−1 Zb1,t−2 ηt−1 ηt−2

Zb1t 0.04 0.86 0.08 0.01 0.00 η_t -0.51 2.63 -1.75 0.67 0.19 Table 4

Estimated parameters for the VAR(2) model on{(Zb1t, ηt)}

Modelling the dynamics of risk neutral densities using DSFM allows to quan- tify the mechanisms governing risk perceptions from agents acting in a market. Insights are obtained in two directions, concerning the autocovariance

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VAR(2)

const Zb3,t−1 Zb3,t−2 γt−1 γt−2

Zb3t 0.00 0.20 0.27 0.01 -0.02 γ_t 0.00 -1.69 0.68 0.81 0.24 Table 5

Estimated parameters for the VAR(2) model on{(Zb3t, γt)}

structure of {Z^b_t}, i.e. the time behaviour of the RN surfaces and their cross- correlation with the skewness and excess kurtosis from the estimated risk neutral densities, i.e. the relation between the dynamics and shape of the obtained RN surfaces. As seen in tables (4) and (5) the excess kurtosis and skewness from qb_t,T at maturity τ = 18 are determined by the corresponding lagged values of Z^b_t.

Based on the presented methodology it is possible to investigate the dynamics of the risk neutral skewness and excess kurtosis based on statistical inference on {Z^bt}. A natural further step is to perform econometric analysis to investigate the cointegration between the lower dimensional time series and macroeconomic and financial indicators. This could provide deeper insights into the relation between risk assessments from investors acting in a market and the flow of economic information at which they are exposed.

4 Acknowledgements

Financial support from the Deutsche Forschungsgemeinschaft via SFB 649

”Economic Risk” is gratefully acknowledged.

A Assumptions

The results from Theorems (2.1) and (2.2), see Borak et al. (2008) rely on the following assumptions:

(A1) The variables X₁₁,...,X_{J T}, ε₁₁,...,ε_{J T} and Z₁, . . . , Z_T are independent. The process Z_t is allowed to be nonrandom.

(A2) For t = 1, ..., T the variables X1t,...,XJ t are identically distributed, have support [0,1]^d and a density f_t that is bounded from below and above on [0,1]^d, uniformly over t = 1, ..., T.

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(A3) We assume that E[ε_jt] = 0 for t= 1, ..., T and j = 1, ..., J and sup

t=1,...,T,j=1,...,J

Eexp[cε²_jt]<∞ for c >0 small enough.

(A4) The functions ψ_k may depend on the increasing indices T and J and are normed so that ^R_[0,1]dψ_k²(x) dx = 1 for k = 1, . . . , K. Furthermore it holds that sup_x∈[0,1]||ψ(x)||=O(K^1/2)

(A5) The components m₀,...,m_L can be approximated byψ₁, ..., ψ_K, i.e.

δ_K = sup

x∈[0,1]^d

Γ∈Ginf |m(x)−Γψ(x)| →0

for l = 0, ..., L and K → ∞. We denote by Γ^∗ the matrix that fulfills sup

x∈[0,1]^d

|m(x)−Γψ(x)| ≤2δK

(A6) There exist constants 0 < C_L < C_U < ∞ such that all eigenvalues of the random matrix T⁻¹^P^T_t=1ZtZ_t^> lie in the interval [CL, CU] with probability tending to one.

(A7) The minimization (2.2) runs over all values of (Γ, z) with sup

x∈[0,1]

1≤t≤Tmax ||Z_t^>Γψ(x)|| ≤M_T

where MT fulfills max1≤t≤T kZtk ≤ MT/Cm (with probability tending to one) for a constant C_m >sup_x∈[0,1]km(x)k.

(A8) It holds that ρ² = (K +T)M_T²log(J T M_T)/(J T) → 0. The dimension L is fixed.

(A9) Z_t is a martingale difference withE[Z_t|Z₁, . . . , Z_t₁] = 0 and for someC > 0 E[||Z_t||²|Z₁, . . . , Z_t₁] < C (a.s). The matrix E[Z_tZ_t^>] has full rank. The process Zt is independent of X11, ..., XT J and ε11, ..., εT J.

(A10) The functionsm₀, ..., m_Lare linearly independent. In particular, no function is equal to 0.

(A11) It holds that (K^1/2MT +T^1/4)(ρ+δK) =^O(1).

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SFB 649 Discussion Paper Series 2008

For a complete list of Discussion Papers published by the SFB 649, please visit http://sfb649.wiwi.hu-berlin.de.

001 "Testing Monotonicity of Pricing Kernels" by Yuri Golubev, Wolfgang Härdle and Roman Timonfeev, January 2008.

002 "Adaptive pointwise estimation in time-inhomogeneous time-series models" by Pavel Cizek, Wolfgang Härdle and Vladimir Spokoiny,

January 2008.

003 "The Bayesian Additive Classification Tree Applied to Credit Risk Modelling" by Junni L. Zhang and Wolfgang Härdle, January 2008.

004 "Independent Component Analysis Via Copula Techniques" by Ray-Bing Chen, Meihui Guo, Wolfgang Härdle and Shih-Feng Huang, January 2008.

005 "The Default Risk of Firms Examined with Smooth Support Vector Machines" by Wolfgang Härdle, Yuh-Jye Lee, Dorothea Schäfer and Yi-Ren Yeh, January 2008.

006 "Value-at-Risk and Expected Shortfall when there is long range dependence" by Wolfgang Härdle and Julius Mungo, Januray 2008.

007 "A Consistent Nonparametric Test for Causality in Quantile" by Kiho Jeong and Wolfgang Härdle, January 2008.

008 "Do Legal Standards Affect Ethical Concerns of Consumers?" by Dirk Engelmann and Dorothea Kübler, January 2008.

009 "Recursive Portfolio Selection with Decision Trees" by Anton Andriyashin, Wolfgang Härdle and Roman Timofeev, January 2008.

010 "Do Public Banks have a Competitive Advantage?" by Astrid Matthey,

January 2008.

011 "Don’t aim too high: the potential costs of high aspirations" by Astrid Matthey and Nadja Dwenger, January 2008.

012 "Visualizing exploratory factor analysis models" by Sigbert Klinke and Cornelia Wagner, January 2008.

013 "House Prices and Replacement Cost: A Micro-Level Analysis" by Rainer Schulz and Axel Werwatz, January 2008.

014 "Support Vector Regression Based GARCH Model with Application to Forecasting Volatility of Financial Returns" by Shiyi Chen, Kiho Jeong and Wolfgang Härdle, January 2008.

015 "Structural Constant Conditional Correlation" by Enzo Weber, January 2008.

016 "Estimating Investment Equations in Imperfect Capital Markets" by Silke Hüttel, Oliver Mußhoff, Martin Odening and Nataliya Zinych, January 2008.

017 "Adaptive Forecasting of the EURIBOR Swap Term Structure" by Oliver Blaskowitz and Helmut Herwatz, January 2008.

018 "Solving, Estimating and Selecting Nonlinear Dynamic Models without the Curse of Dimensionality" by Viktor Winschel and Markus Krätzig,

February 2008.

019 "The Accuracy of Long-term Real Estate Valuations" by Rainer Schulz, Markus Staiber, Martin Wersing and Axel Werwatz, February 2008.

020 "The Impact of International Outsourcing on Labour Market Dynamics in Germany" by Ronald Bachmann and Sebastian Braun, February 2008.

021 "Preferences for Collective versus Individualised Wage Setting" by Tito Boeri and Michael C. Burda, February 2008.

SFB 649, Spandauer Straße 1, D-10178 Berlin http://sfb649.wiwi.hu-berlin.de

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SFB 649, Spandauer Straße 1, D-10178 Berlin http://sfb649.wiwi.hu-berlin.de

022 "Lumpy Labor Adjustment as a Propagation Mechanism of Business Cycles" by Fang Yao, February 2008.

023 "Family Management, Family Ownership and Downsizing: Evidence from S&P 500 Firms" by Jörn Hendrich Block, February 2008.

024 "Skill Specific Unemployment with Imperfect Substitution of Skills" by Runli Xie, March 2008.

025 "Price Adjustment to News with Uncertain Precision" by Nikolaus Hautsch, Dieter Hess and Christoph Müller, March 2008.

026 "Information and Beliefs in a Repeated Normal-form Game" by Dietmar Fehr, Dorothea Kübler and David Danz, March 2008.

027 "The Stochastic Fluctuation of the Quantile Regression Curve" by Wolfgang Härdle and Song Song, March 2008.

028 "Are stewardship and valuation usefulness compatible or alternative objectives of financial accounting?" by Joachim Gassen, March 2008.

029 "Genetic Codes of Mergers, Post Merger Technology Evolution and Why Mergers Fail" by Alexander Cuntz, April 2008.

030 "Using R, LaTeX and Wiki for an Arabic e-learning platform" by Taleb Ahmad, Wolfgang Härdle, Sigbert Klinke and Shafeeqah Al Awadhi, April 2008.

031 "Beyond the business cycle – factors driving aggregate mortality rates"

by Katja Hanewald, April 2008.

032 "Against All Odds? National Sentiment and Wagering on European Football" by Sebastian Braun and Michael Kvasnicka, April 2008.

033 "Are CEOs in Family Firms Paid Like Bureaucrats? Evidence from Bayesian and Frequentist Analyses" by Jörn Hendrich Block, April 2008.

034 "JBendge: An Object-Oriented System for Solving, Estimating and Selecting Nonlinear Dynamic Models" by Viktor Winschel and Markus Krätzig, April 2008.

035 "Stock Picking via Nonsymmetrically Pruned Binary Decision Trees" by Anton Andriyashin, May 2008.

036 "Expected Inflation, Expected Stock Returns, and Money Illusion: What can we learn from Survey Expectations?" by Maik Schmeling and Andreas Schrimpf, May 2008.

037 "The Impact of Individual Investment Behavior for Retirement Welfare:

Evidence from the United States and Germany" by Thomas Post, Helmut Gründl, Joan T. Schmit and Anja Zimmer, May 2008.

038 "Dynamic Semiparametric Factor Models in Risk Neutral Density Estimation" by Enzo Giacomini, Wolfgang Härdle and Volker Krätschmer,

May 2008.